Journal of Commutative Algebra

Cofiniteness and non-vanishing of local cohomology modules

Iraj Bagheriyeh, Kamal Bahmanpour, and Jafar A'Zami

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Let $R$ be a commutative Noetherian local ring, $I$ an ideal of $R$, and let $M$ be a non-zero finitely generated $R$-module. In this paper, we establish some new properties of the local cohomology modules $H^i_I(M)$, $i\geq 0$. In particular, we show that if $(R,\mathfrak{m})$ is a Noetherian local integral domain of dimension $d \leq 4$ which is a homomorphic image of a Cohen-Macaulay ring and $x_1,\ldots,x_n$ is a part of a system of parameters for $R$, then for all $i\geq0$, the $R$-modules $H^i_{I}(R)$ are $I$-cofinite, where $I=(x_1,\ldots,x_n)$. Also, we prove that if $(R,\mathfrak{m})$ is a Noetherian local ring of dimension~$d$ and $x_1,\ldots,x_t$ is a part of a system of parameters for $R$, then $H^{d-t}_{\mathfrak{m}}(H^t_{(x_1,\ldots,x_t)}(R))\neq 0$. In particular, $\mu^{d-t}(\mathfrak{m},H^t_{(x_1,\ldots,x_t)}(R))\neq 0$ and ${\rm injdim}_R(H^t_{(x_1,\ldots,x_t)}(R))\geq d-t$.

Article information

J. Commut. Algebra Volume 6, Number 3 (2014), 305-321.

First available in Project Euclid: 17 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D45: Local cohomology [See also 14B15] 14B15: Local cohomology [See also 13D45, 32C36]
Secondary: 13E05: Noetherian rings and modules

Asso ciated primes cofinite mo dules Krull dimension lo cal cohomology


Bagheriyeh, Iraj; Bahmanpour, Kamal; A'Zami, Jafar. Cofiniteness and non-vanishing of local cohomology modules. J. Commut. Algebra 6 (2014), no. 3, 305--321. doi:10.1216/JCA-2014-6-3-305.

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  • R. Abazari and K. Bahmanpour, Cofiniteness of extension functors of cofinite modules, J. Algebra 330 (2011), 507–516.
  • I. Bagheriyeh, J. A'zami and K. Bahmanpour, Generalization of the Lichtenbaum-Hartshorne vanishing theorem, Comm. Algebra 40 (2012), 134–137.
  • K. Bahmanpour and R. Naghipour, Cofiniteness of local cohomology modules for ideals of small dimension, J. Algebra 321 (2009), 1997–2011.
  • M.P. Brodmann and R.Y. Sharp, Local cohomology; An algebraic introduction with geometric applications, Cambridge University Press, Cambridge, 1998.
  • D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Alg. 121 (1997), 45–52.
  • A. Grothendieck, Local cohomology, Lect. Notes Math. 862, Springer, New York, 1966.
  • R. Hartshorne, Affine duality and cofiniteness, Inv. Math. 9 (1970), 145–164.
  • C. Huneke, Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 2 (1992), 93–108.
  • K.-I. Kawasaki, On the finiteness of Bass numbers of local cohomology modules, Proc. Amer. Math. Soc. 124 (1996), 3275–3279.
  • R. Lü and Z. Tang, The $f$-depth of an ideal on a module, Proc. Amer. Math. Soc. 130 (2001), 1905–1912.
  • T. Marley, Finitely graded local cohomology and depths of graded algebra, Proc. Amer. Math. Soc. 123 (1995), 3601–3607.
  • ––––, The associated primes of local cohomology modules over rings of small dimension, Manuscr. Math. 104 (2001), 519–525.
  • T. Marley and J.C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra 256 (2002), 180–193.
  • H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, UK, 1986.
  • L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambr. Phil. Soc. 107 (1990), 267–271.
  • L. Melkersson, Properties of cofinite modules and application to local cohomology, Math. Proc. Cambr. Phil. Soc. 125 (1999), 417–423.
  • T. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), 649–668.
  • P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), 161–180.
  • K.I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179–191.